Gas resonance device

ABSTRACT

A thermally driven gas resonance device includes a resonance tube (3) which expands in cross-section along its length from one end to the other, a heat source (2) located at the one end of the resonance tube, and an igniter (14) to trigger oscillations in a gas in the tube. The heat source (2) is preferably a pulsed heat source having a repetition frequency corresponding to a resonant frequency of the gas tube (3). The mechanical energy produced in the oscillating gas may be used to operate a pressure swing gas separator by including a bed (16) of molecular sieve material in the other end of the tube (3). Alternatively, the mechanical energy may be used to drive a heat pump (19). In this case a heat sink (21) is located at the other end of the tube (3), a regenerator (20) is also located adjacent the other end, and ports (8) on the side of the regenerator (20) towards the heat source (2) effect heat exchange between the gas in the resonance tube (3) and a source of low grade heat.

This is a continuation of application Ser. No. 07/398,306 filed Aug. 25,1989, now U.S. Pat. No. 4,948,360 which is a continuation of applicationSer. No. 07/117,783, filed Nov. 6, 1987, abandoned.

In an oscillating column of gas a small region of the gas is initiallydisplaced in one direction, is compressed, moves back in the oppositedirection, and expands. During compression the gas is heated and, duringexpansion, is cooled. When such an oscillating column of gas is broughtinto contact with a stationary solid medium, heat transfer takes placebetween the gas and the medium. When the medium has a high effectiveheat capacity compared with that of the gas and a low thermalconductivity in the direction of advancement of oscillations of the gas,it stores heat acquired as a result of the adiabatic compression of thegas and then, returns this stored heat to the gas after its expansion.Whilst this is true for regions of gas which are always located adjacentthe medium a different situation exists at the ends of the medium. Atthe downstream end of the medium, when considered in the direction ofadvancement of the oscillation, a region of gas which is in thermalcontact with the ends of the medium is moved in the one direction awayfrom the medium and compressed during oscillation. The gas is heatedupon compression. Upon subsequently moving in the other direction andexpanding it returns to its position adjacent the end of the medium.Here, since it cools during expansion, it once again accepts heat fromthe medium. This gives rise to a region of heated gas downstream fromthe downstream end of the medium. Conversely, at the upstream end of themedium particles of gas which are not normally in contact with themedium move forward in the one direction during the oscillation arecompressed and heated and then in their forwards position are in thermalcontact with the medium so giving heat to the medium. As this region ofgas moves backwards in the opposite direction to return to its initialposition, it expands and cools. Since in their initial position theparticles of gas are out of thermal contact with the medium this givesrise to a cold region upstream from the upstream end of the medium.

Such a medium located in an oscillating gas column is usually referredto as a regenerator and is often used with Stirling cycle engines.Typically such a regenerator must have as large a surface area aspossible, a high effective heat capacity compared with that of the gasand a low thermal conductivity along the direction of gas motion.Conventionally pads of randomly close-packed metallic wire have beenused as a regenerator but it is also possible to use closely packedstacks of non-metallic plates and these are more efficient with regardto gas friction losses and heat transfer. Thus, the use of a regeneratorenables a temperature difference to be established from an oscillatinggas flow. Conversely, it is also known that if a temperature differenceof sufficient magnitude is applied across such a regeneratoroscillations are spontaneously induced in gas surrounding such aregenerator.

It is also known that oscillations can be established in a column of gaslocated in a resonance chamber by simply applying heat to one end of thechamber if a sufficiently high temperature differential is established.As examples of this gas in an organ pipe can be made to resonate by ahydrogen flame in the base of the pipe as described by Higgins as longago as 1777, and the Taconis oscillations reported in 1949 experiencedwhen placing a tube at room temperature into a cryogenic storage vessel.

A reasoned discussion of these effects is given in an article byWheatley, Hofler, Swift and Migliori entitled "An intrinsicallyirreversible thermoacoustic engine" published in the American Journal ofPhysics Volume 53 (2) February 1985, at page 147.

According to this invention a thermally driven gas resonance devicecomprises a resonance tube which expands in cross-section along itslength from one end to the other, a heat source located at the one endof the resonance tube, and means to trigger oscillations in a gas in theresonance tube.

The heat source may be formed by a simple, indirect heater in which thesource of the heat such as an electrical heating element or a gas or oilburner assembly is used to heat a plate forming or located in the oneend of the gas resonance tube. Preferably the heated plate is finned toimprove the heat transfer from it to the gas at the one end of theresonance tube. A regenerator may be located in the resonance tube closeto but out of contact with the heated plate and from the means totrigger the oscillations. The regenerator consists of a material havinga large surface area, a high effective heat capacity compared with thatof the gas in the resonance tube and a low thermal conductivity alongthe length of the resonance tube, the arrangement being such that, inuse, the heat source sets up a temperature gradient along theregenerator which triggers the oscillations of the gas in the resonancetube.

However, it is very much preferred that the heat source and the means totrigger the oscillations in the gas in the resonance tube are bothformed by a pulsed heat source having a pulse repetition frequencycorresponding to a resonant frequency of the gas resonance tube. Such apulsed heat source may comprise a pulsed combustor or a resonant flamefed with a premixed supply of inflammable gas or vapour and air througha valve, followed by a flame trap, and an ignitor intially to ignite themixture in the one end of the resonance tube or in a combustion chamberleading into the resonance tube. Preferably the valve to admit themixture is formed by a tuned non-return valve which, in response to thepulsed combustion, oscillates between its open and closed states toadmit bursts of mixture into the one end of the resonance tube orcombustion chamber for subsequent ignition. The use of the pulsed heatsource to trigger the oscillations in the gas in the resonance tubeprovides easy starting under wide range of conditions, followed bystable resonant operation. The ignitor may be formed by a sparking plugto cause initial ignition of the pulsed heat source but, once ignitedthe pulsed heat source is preferably self-sustaining. This may be as aresult of subsequent bursts of mixture being ignited by the fading flamefrom a preceding combustion pulse, by spontaneous ignition as a resultof a compression wave or by the ignitor having the form of a glow plugwhich provides a local hot spot to cause ignition.

The pulsed heat source may also include an indirect heater located atthe one end of the resonance tube. The indirect heater may be formed bya heat exchange surface heated by the pulsed heat source to spread theheat of combusion substantially uniformly over the cross-sectional areaof the one end of the resonance tube. Preferably when the gas resonancedevice includes a pulsed combustor the one end of the resonance tube isformed as a parabolic reflector which spreads the effect of the pulsedcombustion more uniformly over the one end of the resonance tube. Inthis case the pulsed combustion is arranged to take place substantiallyat the focus of the parabolic reflector. When the gas resonance deviceincludes a pulsed heat source it may also include a regenerator whichco-operates with a temperature gradient subsisting across it to amplifythe oscillations induced by the pulsed heat source.

With all of these arrangements to generate oscillations in the gas it isnecessary to arrange the shape of the resonance tube both to provide therequired relative pressure and adiabatic temperature amplitudes of thetwo ends of the tube and to minimise gas wall friction losses which tendto inhibit the resonant oscillations of the gas. By having the resonancetube expanding in cross-section from its one end to its other largerpressure and adiabatic amplitudes are developed at the small relative tolarge end and this is discussed in detail dynamical subsequently.Preferably the resonance tube is generally frustoconical in shape withthe ratio of base diameter to height approximately equal to 3:1. Firstlythis provides a diameter to length ratio for the longitudinaloscillation which can be thought of as a gas piston to be as large aspractical thereby minimising wall friction losses. The resonantfrequency of the resonance tube depends mainly upon its length and isindependent of its shape. By making the resonance tube increase incross-sectional area from its one end to its other end it is possible toincrease the mass of gas which oscillates and thereby decrease itsvelocity for a given volume compression ratio. Friction losses areproportional to the cube of the gas velocity and consequently thisreduces the friction losses considerably to enhance the performance ofthe resonance device. Preferably the resonance tube has a frusto-ogivalshape in longitudinal-section so that, when seen in cross-section, itsside walls are curved. This provides a further increase in the mass ofoscillating gas closer to the one end and so enhances the reduction infriction losses still further.

The mechanical energy produced in the oscillating gas in the gasresonance device may be used to operate a pressure swing gas separatorwith a molecular sieve material. One of the most straightforwardarrangements is to use the gas resonance device in an apparatus for thepressure swing separation of oxygen from air. In this case the other endof the resonance tube contains a molecular sieve material, a gasexchange port is provided on the side of the molecular sieve materialtowards the heat source, and a gas outlet is provided upon the side ofthe molecular sieve material remote from the heat source. Duringoscillation as air moves forwards through the bed of molecular sievematerial nitrogen is preferentially adsorbed by the molecular sievematerial. As the air moves backwards a reduced pressure is created andgases adsorbed onto the surface of the molecular sieve material aredesorbed. Thus, when the molecular sieve material is subjected to theoscillations generated in the resonance tube nitrogen, which ispreferentially adsorbed by the molecular sieve material tends to returnto the inside of the resonance tube and hence out of the gas exchangeport, whereas oxygen, which is less adsorbed by the molecular sievematerial, tends to be driven through the bed of the molecular sievematerial and out of the gas outlet at the downstream side of themolecular sieve material. The finite displacements of the gas that occurduring oscillation create a mean pressure slightly above ambient in theresonance tube so that a continuous flow of separated oxygen emergesbelow the bed of molecular sieve material.

Typically the molecular sieve material is an expanded zeolite but activecarbon may also be used. The molecular sieve material preferably hassufficient surface area to permit a high nitrogen adsorbtion rate and ithas been found that the cumulative rate of adsorbtion and desorbtion isproportional to pressure swing and nearly independent of cycle rate.

In an alternative configuration the mechanical energy produced in theoscillating gas in the resonance tube is used to drive a heat pump. Inthis case the gas resonance device includes a heat sink located at itsother end, a regenerator located adjacent the other end, and means onthe side of the regenerator towards the heat source to effect heatexchange between the gas in the resonance tube and a source of low gradeheat.

With this arrangement the effects discussed earlier are used to providea heat engine driven heat pump. Thus the oscillations in the gas in theresonance tube are applied to the regenerator to produce a temperaturedifferential across it with the gas downstream of the regenerator at theother end of the resonance tube being heated and with the gas upstreamfrom the regenerator being cooled. The heat exchange that takes placeupstream of the regenerator provides the heat for the expansion of thegas upstream from the regenerator and provides the source of the heatwhich is pumped to provide part of the heat removed by the heat sink atthe other end of the resonance tube. In addition to this the heat sinkat the other end of the resonance tube also receives heat provided bythe heat source. The applicant has coined the acronym HASER to describethis type of heat engine driven heat pump with the acronym standing for"Heat Amplification by Stimulated Emission of Radiation" by analogy withthe acronyms laser and maser.

When the source of low grade heat is the atmosphere, it is preferredthat a direct heat exchange takes place between the atmosphere and gasin a region upstream of the regenerator. To provide this gas exchangeports are provided in the wall of the resonance tube at the position ofa pressure null point. As the longitudinal vibrations pass down theresonance tube the atmosphere tends to be drawn into the resonance tubethrough the ports after the compression oscillation has passed theports. The gas that is drawn into the resonance tube from the atmospherethen mixes with the gas in the resonance tube with a resulting heatexchange taking place between the gas from the atmosphere and the gasalready in the resonance tube. The next oscillation then tends to drivethe now cooled atmospheric air out of the ports.

Preferably however the haser also includes a fan to drive air from theatmosphere through the gas exchange ports into the resonance tube.Preferably an outer chamber surrounds the resonance tube with the fanlocated at the top, that is the end of the resonance tube with the heatsource, and a corrugated annular baffle adjacent the gas exchange portsto direct air blown by the fan through half of the ports and allowcooled air to leave from the other half of the ports and flow throughthe lower portion of the outer chamber. The air flowing through theouter chamber absorbs heat given off from the heat source and upper partof the gas resonance tube and this heat is re-introduced into the systemas part of the low grade heat so further improving the heat output ofthe haser.

The expansion of the cross-section of the resonance tube from the oneend of the other has further advantages in a haser. The relativecross-sectional areas of the two ends determine the compression ratiodeveloped at them. A small cross-section leads to a high compressionratio and vice versa. The effect of this can be derived from acoustictheory of small displacements and is developed for the particularexample subsequently. The expanding cross-section of the resonance tubefrom the one end to the other leads to a high compression ratio for thedriving end and a low compression ratio at the pump end and thisprovides the optimum thermal efficiency.

The heat sink at the other end of the resonance tube may comprise ashallow pool of water and, in this case, it is preferred that fins of agood thermal conductor such as metal are in thermal contact with thepool of water and extend in the space between the pool of water and thedownstream side of the regenerator. Such a heat sink has a good thermalcontact with the hot gas downstream of the regenerator. The water in thepool is circulated around a system to carry the heat away from the otherend of the resonance tube and this circulation system may includenon-return valves on both sides of the pool so that the water is drivenaround the system by the pressure fluctuations inside the resonance tubeacting on the surface of the water in the pool.

Such a haser has particular application as a hot water generator for usein heating and cooling a residential building. The haser is typicallylocated in the roof space of a building and, in winter, the roof spaceis ventilated or air from outside ducted to it so that air provides thesource of low grade heat. The heat sink at the other end of theresonance tube is used to heat water to a temperature of say 40° C. andthis water is used for domestic hot water requirements and is circulatedaround a central heating system of the building. During summer the haseris used to provide cooling for the building by closing the ventilationof the roof space and opening cooling vents in ceilings of the roomsbelow the roof space or ducting the air leaving the haser to the rooms.Water from the heat sink at the other end of the resonance chamber isused for domestic hot water requirements and also is led away to a heatexchanger outside the building where it is cooled. The resulting coolair discharged from the resonance tube cools the roof space and, inturn, through the cooling vents in the ceilings, or via the ductingcools the building.

A pressure swing gas separator may be combined with a haser by placingmolecular sieve material in the resonance cavity above the regenerator.With this combination the output from the outlet ports is cool and richin nitrogen. Such an output is good for preserving perishables and sucha combined device provides a readily portable, self-contained source ofnitrogen enriched cold air.

A particular example of a haser in accordance with this invention willnow be described with reference to the accompanying drawings, in which:

FIG. 1 is a partly sectioned side elevation of a pressure swing gasseparator;

FIG. 2 is a partly sectioned side elevation of a haser;

FIG. 3 is a cross-section through a heat source;

FIGS. 4a and 4b are diagrams illustrating the dimensions of theresonance tube and the gas displacements for parallel and conical tubes,respectively;

FIG. 5 is a graph showing the characteristics of the resonance tube;

FIG. 6 is another graph illustrating how displacement and densityamplitudes vary with respect to time over the length of the resonancetube;

FIG. 7 is a further graph illustrating the effect of ogival correction;and,

FIG. 8 is a temperature against position diagram to illustrate theoperation of the regenerator.

Both the pressure swing gas separator shown in FIG. 1 and the hasershown in FIG. 2 include a heat engine 1 formed by a pulsed heat source 2mounted at one end of a resonance tube 3 which is ogival in longitudinalsection. The overall dimensions of the resonance tube 3 are such thatits height is about three times its base diameter. A regenerator 4 maybe included towards the top of the resonance tube and this is made froma non-metallic honeycomb which is typically made from glass or aglass-like material. An outer concentric annular chamber 5 surrounds theresonance tube 3 and an electrically driven fan 6 is mounted at the topto blow air downwards through the chamber 5. A corrugated annular baffle7 directs the flow of air through alternate open ports 8 provided in theside wall of the resonance tube 3 at a pressure null point. Air isdischarged through the other ports 8 and a lower portion of the outerchamber 5 The open ports 8 produce orifice flow and therefore inwardsair flow through alternate ports 8 is strongly converging which ensuresthat charge and discharge through the ports 8 is not unduly mixed.

The pulsed heat source 2 is shown in more detail in FIG. 3 and comprisesa gas mixing space 9 to which gas and air are supplied and in which theyare mixed, a resonant non-return valve 10 of similar resonant frequencyto that of the resonance tube 3, and a flame trap 11. The resonancenon-return valve 10 may be similar to those fitted to two-stroke enginesand comprise an open port 12 covered by a springy plate 13 which isfixed along one edge to the port 12. In response to the instantaneouspressure in the resonance tube 3 being greater than that in the gasmixing space 9 the valve is held closed with the springy plate 13forming a seal against the edges of the port 12, and in response to aninstantaneous reduction in pressure in the resonance tube 3 with respectto that in the gas mixing space 9, the springy plate 13 bends to allowthe gas and air mixture to pass through the port 12 and into theresonance tube 3.

In a preferred configuration which leads to gas mixture delivery moreclosely in phase with the resonance chamber compression pulse, and thusto concomitant improvement in pulsed combustion, the resonancenon-return valve consists of a metal disc of relatively large diameter,placed co-axially with the combustion chamber, clamped at its edges to aslightly concave bedplate in which the flame trap is centrally located.Gas mixture is introduced at low pressure to an internal annulus closeto the clamped edges, and is thereby fed radially inwards in pulsestowards the flame trap. The disc is of such thickness that its naturalfrequency of axial oscillation is lower than that of the resonancecavity so that the combined effect of the gas damping and the cavitypressure pulses is to produce substantially antiphase oscillations ofthe disc at the resonant frequency of the cavity. These oscillationsintroduce gas mixture through the flame trap to the combustion chamberat the time of pressure rise instead of the time of maximum suction, andthe former diminishes the extent of premature combustion, which isinefficient with regard to heat engine operation.

The pulsed heat source 2 also includes a sparking plug 14 and the top ofthe resonance tube 3 is formed as a parabolic reflector 15 which spreadsthe effect of the pulsed heat source substantially uniformly over theend of the resonance tube 3.

The heat engine 1 drives a gas oscillation down the resonance tube 3 andthe vertically oscillating mass of gas functions as a piston producingpressure and adiabatic temperature fluctuations at top and bottom of thetube 3. The oscillations are triggered by the sparking plug 14 initiallyigniting the gas and air mixture introduced into the top of theresonance tube 3 and then, as the gas in the tube 3 begins to resonateand the valve 10 introduces successive bursts of mixture these areignited by the fading flame from the previous ignition. This produces apulsed combustion which, in a device having a resonance tube of lengthabout 1 m has a repetition frequency of around 200 Hz. The regenerator 4increases the efficiency of the heat engine 1 by increasing thetemperature of the top end of the resonance tube 3 and increasing theamplitude of the ocillations produced.

The heat engine 1 just described may be used to provide the mechanicalenergy input for a pressure swing gas separator and, in this case, asshown in FIG. 1 a shallow bed 16 of a zeolite which preferentiallyadsorbs nitrogen is placed towards the lower end of the resonance tube 3and the base of the resonance tube is closed by a plate 17 including agas outlet 18. During resonant oscillation in the resonance tube 3 asthe air moves forwards into the zeolite bed 16 nitrogen ispreferentially adsorbed by the zeolite. As the air moves backwards areduced pressure is created and the gases adsorbed onto the surface ofthe zeolite are desorbed so that air rich in nitrogen is desorbed. As aresult of the finite displacements of gas that occur during oscillationthe average pressure inside the resonance tube 3 is greater thanatmospheric so that a flow of gas passes through the zeolite bed 16resulting in a flow of gas out of the output 18 which is rich in oxygenwhilst the flow of gas out of the ports 8 and through the lower part ofthe chamber 5 is rich in nitrogen.

The heat engine 1 may alternatively be used to provide the mechanicalenergy to drive a heat pump 19. A heat engine driven heat pump has anoverall coefficient of performance (COP) where ##EQU1## in excess ofunity, provided that the adiabatic temperature ratio of the formersignificantly exceeds that of the latter. The heat output may also bedirectly supplemented by heat rejected from the heat engine 1. The heatpump part 19 of the apparatus comprises a regenerator 20 which is madefrom a non-metallic honeycomb which is typically made of glass orglass-like material and a heat sink 21. The heat sink 21 is formed by ashallow pool of water 22 in the large diameter end of the resonance tube3 and metallic fins 23 in thermal contact with the shallow pool of water22 extend into the resonance tube 3 towards the regenerator 20. Air,which in this case provides the low grade source of heat enters andleaves through the ports 8 and heat is extracted from this air by theheat pump 19 and transferred to the water 22 in the heat sink 21.

Thus, in operation, gas oscillations are induced by the heat engine 1inside the resonance tube 3. These oscillations provide the drivingpower for the heat pump 19 including the regenerator 20. As the gasoscillates around the regenerator 20 the space beneath the regenerator20 is heated and the space above the regenerator 20 cooled. Air flowthrough the ports 8 mixes with the gas in the resonance tube 3 and givesheat to the gas in the resonance tube 3 above the regenerator 20. Theheat sink 21 removes the build up of heat beneath the regenerator 20.

Typically the level of the shallow pool 22 of circulating water in theheat-sink 21 is controlled by a float valve (not shown). A water inletand outlet for the pool 22 includes non-return valves (not shown) andthe gas oscillations set up in the resonance tube 3 act on the surfaceof the water in the pool 22 and cause circulation of the water throughthe inlet and outlet non-return valves. Typically the water outlettemperature is about 40° C. and this can be used as a source of domestichot water or a source of hot water for driving a central heating system.Typically a haser as shown in this example is mounted in the roof spaceof a house which is ventilated in the winter to allow air from theatmosphere to provide the source of low grade heat entering and leavingthe ports 8. If desired to cool the building during the summer months,ventilators for the roof space would be closed and ceiling louvresopened to allow the cold air generated by the haser to gravitate intothe house. In this case the hot water discharged from the heat sink 21is, after the needs for domestic hot water have been supplied, passed toan atmospheric heat exchanger out of doors to dissipate the heatgenerated in the haser before being recirculated. The target value ofthe COP for such a haser would be 2 in the heating mode.

The details of the gas resonance dynamics, the desirability of theogival shape of the resonance chamber, a discussion on the wall frictionlosses, and a discussion on the characteristics of the regenerator 20will now be provided.

The Dynamics of Gas Resonance

It is necessary to develop a quantitative treatment of gas motions inorder to design a haser. Linear elastic displacements of a uniform solidor fluid in a parallel configuration (FIG. 4a) are governed by thewell-known equation: ##EQU2## where α is displacement at referencedistance x, t is time lapse and c is velocity of sound. For a standingwave in a tube with closed ends where cross-section is also uniform, αis proportional to: ##EQU3## The corresponding governing equation forspherical symmetry, applicable to oscillatory flow in a truncated cone,has been known since the times of Cauchy and Poisson, and is: ##EQU4##where r is defined as in FIG. 4b. In both cases ##EQU5## where γ is theratio of specific heat and p_(o) is the mean pressure at which densityis p_(o). When the cone is truncated at the radii b and a it may readilybe verified that a standing wave solution with arbitrary constant A is:##EQU6## so that: ##EQU7## which becomes zero both at r=b, and r=a sincea+b=1. It is convenient in the following to put: ##EQU8## θ' is amaximum when ##EQU9## hence when θ=tan (θ-θ_(o)), which can be solved interms of θ_(o) for particular values of θ, as in FIG. 5, noting that##EQU10## where d is the distance of the maximum position from the smallend.

Neglecting second order small quantities it may be shown that theinstantaneous density ratio is: ##EQU11## which is unity when θ=-tan(θ-θ_(o)), and is solved in the same way to calculate ##EQU12## withresults plotted in FIG. 5. It is seen that maximum displacement andvelocity occur at a position displaced from the midpoint towards thesmall end of the cone, and the density and pressure null point isdisplaced by a corresponding distance towards the large end.

The extreme values of ρ_(o) from equation 5 are obtained when cos ωt andcos (θ-θ_(o)) are both ±1, so that volume compression ratios m_(o) andm_(l) may be defined, relating to the small and large ends respectively.Then: ##EQU13## If m_(o) and ##EQU14## are specified, A may beeliminated by the deduction from equation 6 that: ##EQU15## Furthermore,it may be deduced frOm equations 3 or 4, in combination with FIG. 5, forany point distance x from the small end that: ##EQU16## In particular,x=d for maximum displacement or velocity, and x=d' for the correspondingdisplacement or velocity at the null point of density or pressurechange. However, when taking into account the finite displacements as inthe plot of extreme values of ##EQU17## against ##EQU18## shown in FIG.6, it is seen that the actual value of ##EQU19## at the null point isproportionally reduced relative to the unit reference value. The valueof this proportional reduction ##EQU20## may be calculated from theslope of the curves derived from equation 5, referred to the null pointand multiplied by the relevant displacement. Thus: ##EQU21## obtainedfrom equation 8 with x=d'. It may be noted from this that there is anequivalent excess mean density and pressure for open cycle applications,since inflow and outflow must be balanced.

The preceding analysis is based as stated upon linear-elasticrelationships, which are realised with gases only for smalldisplacements. Adiabatic behaviour is non-linear, but the effects of thenon-linearity have been extensively studied for free pistonapplications, and found to be significant only for high compressionratios, which are themselves marginally increased from values calculatedby linear elastic methods, and time rates of change are momentarilyincreased, so that frequencies are somewhat higher than calculated. Thelinear theory successfully predicts the location of the pressure nullpoint in experiments with the oscillating haser, but observed resonantfrequencies are higher than predicted values, consistent both with theaforementioned non-linearity of adiabatic gas compression, and theincrease of sonic velocity with temperature in the upper part of theresonance tube. It may therefore be claimed that the theory as presentedis sufficiently accurate to be used with confidence, and thatcorresponding proportional adiabatic excursions of pressure and absolutetemperature may be obtained from the proportional density excursions byraising the latter to the powers γ and γ-1, respectively.

Coefficient of Performance

Since the Carnot thermal efficiency at the heated end, determined fromthe ratio of absolute temperatures in the cycle, is: ##EQU22## and thecorresponding heat pump gain, determined from the ratio of absolutetemperatures in the cycle at the absorber end, is: ##EQU23## it followsthat the idealised coefficient of performance is: ##EQU24## With γ-1.4for air, this value for the example of FIG. 6 is 2.77, but it will bemodified by several factors.

Firstly, it can be argued that heat is not rejected at the lowtemperature corresponding with expansion at the heated end, nor is itabsorbed at a temperature corresponding with full compression at theabsorber end, This is considered later with reference to theregenerators where it is demonstrated that the function of theregenerators is to lift the average temperatures at both ends, so thatthe ideal is more closely approached.

Secondly, although a proportion of the wall friction heat could berecovered with a water jacket, as in the case with heat rejected at theheat engine end, the former has to be provided as mechanical powersubject to the limited thermal efficiency of the heat engine function,and is subject thereby to substantial leverage. Mechanical power is alsodissipated in the air entry and discharge through the ports 7 and 8 buta proportion of this is common to wall friction loss, since both areconcerned with the boundary layer.

The Ogival Shape Modification

The purpose of this for given end diameters is to increase the mass ofoscillating gas, thereby to decrease its velocity for given volumecompression ratios. The effect is worth incorporation since frictionloss is proportional to the cube of velocity. A full numerical analysiswould be feasible, but the effect is likely to be contained adequatelywithin a correction.

The key to the correction procedure is found in the first orderdependence of natural frequency f on length 1, irrespective of shape,whether parallel, conical or ogival, such that: ##EQU25## Angularfrequency ω for an oscillating system is also dependent upon: ##EQU26##where k is stiffness and m is mass so that, in the ogival case,stiffness must be considered to be increased by the same proportion asmass. The consequence of this is seen in FIG. 7, where there are threenotional concentric cones OA, OB and OC. OA is that which contains thediameters of the two ends, and OB circumscribes the ogival curve. OC isan interpolation. It will be recognised that the preceding theory isunchanged in relation to the three cones, since all have the same ratio##EQU27## Cone OC intercepts the ogival curve to define a mass zone andtwo stiffness zones. If the diameter ratio between OA and OC is 1+e, itfollows from considerations of volume and cross-section area that thestiffnesses are also increased by the ratio 1+e. However, irrespectiveof the definition of e the enclosed volume designated as mass isincreased by a ratio in excess of 1+2e, and this anomaly can beexplained in terms of the hydrodynamic concept of virtual mass, which inthis case is negative because of velocities which are lower within theenlarged cross-section. Since stiffness can be calculated withoutambiguity it is appropriate to consider that both stiffness and apparentmass are increased by 1+e, and that OC is defined as the neutral axis ofthe ogival curve between the two ends (equal surface area), such that##EQU28## Then the ogival shape behaves in relation to friction loss butnot to compression ratio or output as though it is a cone with diametersincreased in the ratio 1+e, but with peak velocities decreased in theratio ##EQU29## The effect of the ogival shape shown in FIGS. 1 and 2 isto reduce friction loss by 50% for circumstances which are otherwiseequivalent.

Wall Friction Losses

In the absence of specific data for oscillatory flow in cones, it isappropriate to consider steady state boundary layer friction in aparallel pipe of equivalent length and diameter. The frictioncoefficient c_(f) depends upon Reynold's number and proportional surfaceroughness, and is given by the standard data of Nikuradse. For theReynold's numbers >10⁶ and surface roughness <50 micron, a suitablevalue for c_(f) would be 0.0035.

If the axial displacement in a cylindrical tube of radius R and halfwavelength 1 is given by: ##EQU30## the axial velocity ##EQU31## thewall shear stress ##EQU32## and the local instantaneous work rate##EQU33## The average work rate W, integrated over the length of thetube and with respect to time, is: ##EQU34## Noting that the bracketedquantity is ##EQU35## and evaluating the Gamma functions. ##EQU36##u_(o) is the amplitude of axial velocity and A is total area of wall.

Expressed as a fraction η of the maximum kinetic energy of the gas inthe tube, the accumulation of friction work over a half stroke is:##EQU37## These considerations show that, although wall friction lossesare a significant fraction of the developed mechanical power, they arecontainable particularly if the ogival shape is incorporated.

Characteristics of Regenerators

Regenerators have been used since Stirling cycle engines were firstconstructed, and it is accepted that they confer large increases ofefficiency. Their function as thermal filters is to store heat acquiredat one part of a reciprocating cycle as gas is passed during anadiabatic volume change, and to return it to the gas as the cycle isreversed. Their two essential properties are thus a large surface areaexposed to the gas, coupled with a small thermal conductivity in thedirection of gas motion. Typically they have the form of stacks ofspaced non-metallic plates since these are efficient with regard to gasfriction losses and heat transfer.

The beneficial use of regeneration in the present case may be seen froma plot of extreme values of adiabatic temperature change (FIG. 7)developed from the corresponding plot of density ratios in FIG. 5. Thesloping lines in FIG. 7 quantitatively depict the paths taken by packetsof gas undergoing adiabatic temperature changes, but in a reversiblemanner for simplicity, since in practice the lines would be loopsacounting for heat transfer. The positions of regenerators 3 and 4 areshown, and it may be noted that these do not extend to the ends of theresonance tube where the heat transfer surfaces are placed. The gapsbetween are sufficient to avoid thermal contact.

The lines BB' and EE' represent gas packets which always remain withinthe regenerators. These packets pump heat against a temperature gradientby acquiring it when expanded and cold at positions towards the centreof the cavity, and releasing it by heat exchange when compressed and hotat positions towards the end of the cavity. Thus, there are temperaturegradients within the regenerators, which slope up towards the ends ofthe cavity. The mean temperatures of the regenerators are also aboveambient because heat exchange from compressed gas is more effective thanfrom rarefied gas.

The most profound effect of the regenerators occurs at their ends nearto the ends of the cavity, because in these positions gas enters whenexpanded and cold and receives heat. It leaves as it becomes compressedand hot, so that the mean temperature at the ends of the cavity issignificantly raised. The effect may be traced with lines AA' and FF'.The reverse effect occurs at the inner ends of the regenerators, as seenfrom lines CC' and DD'. The overall effect is that mean temperatures aresignificantly raised by as much as half the temperature amplitudeoutboard of the regenerators, and lowered by a smaller amount over alarger volume inboard of the regenerators.

The regenerator length should exceed the gross displacement at thepreferred location, and the gap between the lower regenerator and theheat absorber should be the practical minimum. An optimal criterion forthe regenerator material is that the conductive heat penetration depthfor each cycle should not exceed the strip thickness, and this is statedfrom the relevant transient heat flow treatment as: ##EQU38## For t=0.1mm and ω=817/sec, this indicates a preferred thermal diffusivity αapproximately equal to 10⁻² cm² /sec, which would be satisfied by glassymaterials. Metals are too conductive.

I claim:
 1. A thermally driven gas resonance device comprising:anelongate resonance tube (3), said resonance tube having an upper portionand a lower portion, said upper portion expanding in cross-section alongits entire length from a first end thereof to a second end thereof; aheat source (2), said heat source being located at said first end ofsaid resonance tube; and, means for triggering oscillations in a gas insaid resonance tube, wherein said resonance tube has a frusto-ogivalshape and constantly expanding in cross-sectioned area along its entirelength such that, when viewed along a longitudinal section thereof, sidewalls of said resonance tube are bowed outwardly over their entirelength.
 2. The gas resonance device of claim 1, wherein said heat sourceand said means to trigger said oscillations in said gas in saidresonance tube are both formed by a pulsed heat source, said pulsed heatsource having a pulse repetition frequency corresponding to a resonantfrequency of said gas resonance tube.
 3. The gas resonance device ofclaim 2, wherein said pulsed heat source also includes an indirectheater located at said one end of said resonance tube.